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          <h1 class="post-title" itemprop="name headline">Decisiong Tree:ID3 C4.5 CART</h1>
        

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        <ol>
<li><p>决策树<br><strong><em>问题：如何挑选用于分裂节点的特征–&gt;ID3 C4.5 …(一个标准：使分裂出来的节点尽可能纯，即一个分支尽可能属于同类)</em></strong></p>
</li>
<li><p>ID3<br><em><strong>信息增益</strong></em></p>
<p> 信息增益 = 信息熵 - 条件熵</p>
<ul>
<li>信息增益：针对每个 <em><strong>属性</strong></em></li>
<li>信息熵：整个样本空间的不确定度。其中Pk一定是label取值的概率。</li>
<li><p>条件熵：给定某个属性，求其信息熵</p>
<p>–&gt; 问题：某属性所包括的类别越多，信息增益越大。极限：每个类别仅有1个实例（label数量为1），log p = log1 = 0， 所以最终条件熵=0。或：属性类别越多，条件熵越小，其纯度越高。</p>
<p>–&gt; 信息增益准则其实是对可取值数目较多的属性有所偏好！</p>
<p>–&gt; 泛化能力不强</p>
</li>
</ul>
</li>
<li><p>C4.5 <em><strong>信息增益率+信息增益</strong></em></p>
<p> 属性a的信息增益率 = 属性a的信息增益 / a的某个固有统计量IV(a)</p>
<p> <img src="https://pic4.zhimg.com/80/v2-812104c0291d20935e910919a9fa5c27_hd.png" alt="IV(a)公式"></p>
<p> V为a的取值数目。<br> （实际上是属性a的信息熵）</p>
<ul>
<li>直接使用信息增益率：偏好取值数目小的属性。</li>
<li>先选择高于平均水平信息增益的属性，再选择最高信息增益率的属性。</li>
</ul>
</li>
<li><p>CART <em><strong>基尼系数+MAE/MSE</strong></em></p>
<p>与ID3、C4.5的不同：形成二叉树，因此 –&gt; 既要确定要分割的属性，也要确定要分割的值</p>
<p>回归树和分类树的区别在于样本输出，如果样本输出是离散值，那么这是一颗分类树。如果果样本输出是连续值，那么那么这是一颗回归树。</p>
<p>回归树和分类树在处理连续特征的时候有区别：</p>
<ul>
<li>回归树：MAE/MSE<ul>
<li>example(MSE)：<blockquote>
<ol>
<li>考虑数据集 D 上的所有特征 j，遍历每一个特征下所有可能的取值或者切分点 s，将数据集 D 划分成两部分 D1 和 D2</li>
<li>分别计算上述两个子集的平方误差和，选择最小的平方误差对应的特征与分割点，生成两个子节点。</li>
<li>对上述两个子节点递归调用步骤1 2,直到满足停止条件。</li>
</ol>
</blockquote>
</li>
</ul>
</li>
<li><p>分类树：(Gini)</p>
<p>$$ Gini(A) = \sum_{k=1}^K p_k(1-p_k)$$<br><img src="https://img-blog.csdn.net/20150109184544578?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvQW5kcm9pZGx1c2hhbmdkZXJlbg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/Center" alt="某属性A的基尼系数"><br>基尼系数越小，纯度越高</p>
</li>
</ul>
<blockquote>
<ol>
<li>对每个特征 A，对它的所有可能取值 a，将数据集分为 A＝a，和 A!＝a 两个子集，计算集合 D 的基尼指数：<br>Gini(A) = D1/D <em> Gini(D1) + D2/D </em> Gini(D2)</li>
<li>遍历所有的特征 A，计算其所有可能取值 a 的基尼指数，选择 D 的基尼指数最小值对应的特征及切分点作为最优的划分，将数据分为两个子集。</li>
<li>对上述两个子节点递归调用步骤1 2, 直到满足停止条件。</li>
<li>生成 CART 决策树。</li>
</ol>
</blockquote>
<ul>
<li>使用基尼系数处理离散特征：不停地二分</li>
<li><p>使用基尼系数处理连续特征，将所有n个取值从大到小排列，得到n-1个切分点，求所有的切分点的基尼系数，取最小的，做到连续特征的离散化<br>  停止条件有：</p>
<ol>
<li>节点中的样本个数小于预定阈值;</li>
<li>样本集的Gini系数小于预定阈值（此时样本基本属于同一类）;</li>
<li>没有更多特征。</li>
</ol>
</li>
<li><p>剪枝</p>
</li>
<li>例子：<a href="https://www.jianshu.com/p/b90a9ce05b28" target="_blank" rel="noopener">example</a></li>
</ul>
</li>
<li><p>控制决策树过拟合的方法</p>
<ul>
<li>剪枝</li>
<li>控制终止条件，避免树形结构过细</li>
<li>构建随机森林</li>
</ul>
</li>
<li><p>剪枝</p>
</li>
</ol>
<ul>
<li>前剪枝：<ul>
<li>在生成树的时候，控制终止条件：最小样本数；样本集的最小基尼系数；树的深度</li>
<li>由于预剪枝不必生成整棵决策树，且算法相对简单，效率很高，适合解决大规模问题。但是尽管这一方法看起来很直接，但是怎样精确地估计何时停止树的增长是相当困难的。</li>
<li>预剪枝有一个缺点， 即视野效果问题 。 也就是说在相同的标准下，也许当前的扩展会造成过度拟合训练数据，但是更进一步的扩展能够满足要求，也有可能准确地拟合训练数据。这将使得算法过早地停止决策树的构造。</li>
</ul>
</li>
<li>后剪枝<ul>
<li>在已生成过拟合决策树上进行剪枝，可以得到简化版的剪枝决策树。</li>
</ul>
</li>
</ul>
<p>1）REP-错误率降低剪枝</p>
<p>对于决策树T的每棵非叶子树S, 用叶子替代这棵子树. 如果S被叶子替代后形成的新树关于D的误差等于或小于S关于D所产生的误差, 则用叶子替代子树S。</p>
<p><strong>优点：</strong></p>
<ul>
<li>REP 是当前最简单的事后剪枝方法之一。</li>
<li>它的计算复杂性是线性的。</li>
<li>和原始决策树相比，修剪后的决策树对未来新事例的预测偏差较小。</li>
</ul>
<p><strong>缺点：</strong></p>
<ul>
<li>但在数据量较少的情况下很少应用. REP方法趋于过拟合( overfitting) , 这是因为训练数据集中存在的特性在剪枝过程中都被忽略了, 当剪枝数据集比训练数据集小得多时 , 这个问题特别值得注意.</li>
</ul>
<p>2）PEP-悲观剪枝</p>
<ul>
<li><p>CCP-代价复杂度剪枝</p>
</li>
<li><p>MEP-最小错误剪枝</p>
</li>
</ul>

      
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  <script type="text/javascript">
    // Popup Window;
    var isfetched = false;
    var isXml = true;
    // Search DB path;
    var search_path = "search.xml";
    if (search_path.length === 0) {
      search_path = "search.xml";
    } else if (/json$/i.test(search_path)) {
      isXml = false;
    }
    var path = "/" + search_path;
    // monitor main search box;

    var onPopupClose = function (e) {
      $('.popup').hide();
      $('#local-search-input').val('');
      $('.search-result-list').remove();
      $('#no-result').remove();
      $(".local-search-pop-overlay").remove();
      $('body').css('overflow', '');
    }

    function proceedsearch() {
      $("body")
        .append('<div class="search-popup-overlay local-search-pop-overlay"></div>')
        .css('overflow', 'hidden');
      $('.search-popup-overlay').click(onPopupClose);
      $('.popup').toggle();
      var $localSearchInput = $('#local-search-input');
      $localSearchInput.attr("autocapitalize", "none");
      $localSearchInput.attr("autocorrect", "off");
      $localSearchInput.focus();
    }

    // search function;
    var searchFunc = function(path, search_id, content_id) {
      'use strict';

      // start loading animation
      $("body")
        .append('<div class="search-popup-overlay local-search-pop-overlay">' +
          '<div id="search-loading-icon">' +
          '<i class="fa fa-spinner fa-pulse fa-5x fa-fw"></i>' +
          '</div>' +
          '</div>')
        .css('overflow', 'hidden');
      $("#search-loading-icon").css('margin', '20% auto 0 auto').css('text-align', 'center');

      $.ajax({
        url: path,
        dataType: isXml ? "xml" : "json",
        async: true,
        success: function(res) {
          // get the contents from search data
          isfetched = true;
          $('.popup').detach().appendTo('.header-inner');
          var datas = isXml ? $("entry", res).map(function() {
            return {
              title: $("title", this).text(),
              content: $("content",this).text(),
              url: $("url" , this).text()
            };
          }).get() : res;
          var input = document.getElementById(search_id);
          var resultContent = document.getElementById(content_id);
          var inputEventFunction = function() {
            var searchText = input.value.trim().toLowerCase();
            var keywords = searchText.split(/[\s\-]+/);
            if (keywords.length > 1) {
              keywords.push(searchText);
            }
            var resultItems = [];
            if (searchText.length > 0) {
              // perform local searching
              datas.forEach(function(data) {
                var isMatch = false;
                var hitCount = 0;
                var searchTextCount = 0;
                var title = data.title.trim();
                var titleInLowerCase = title.toLowerCase();
                var content = data.content.trim().replace(/<[^>]+>/g,"");
                var contentInLowerCase = content.toLowerCase();
                var articleUrl = decodeURIComponent(data.url);
                var indexOfTitle = [];
                var indexOfContent = [];
                // only match articles with not empty titles
                if(title != '') {
                  keywords.forEach(function(keyword) {
                    function getIndexByWord(word, text, caseSensitive) {
                      var wordLen = word.length;
                      if (wordLen === 0) {
                        return [];
                      }
                      var startPosition = 0, position = [], index = [];
                      if (!caseSensitive) {
                        text = text.toLowerCase();
                        word = word.toLowerCase();
                      }
                      while ((position = text.indexOf(word, startPosition)) > -1) {
                        index.push({position: position, word: word});
                        startPosition = position + wordLen;
                      }
                      return index;
                    }

                    indexOfTitle = indexOfTitle.concat(getIndexByWord(keyword, titleInLowerCase, false));
                    indexOfContent = indexOfContent.concat(getIndexByWord(keyword, contentInLowerCase, false));
                  });
                  if (indexOfTitle.length > 0 || indexOfContent.length > 0) {
                    isMatch = true;
                    hitCount = indexOfTitle.length + indexOfContent.length;
                  }
                }

                // show search results

                if (isMatch) {
                  // sort index by position of keyword

                  [indexOfTitle, indexOfContent].forEach(function (index) {
                    index.sort(function (itemLeft, itemRight) {
                      if (itemRight.position !== itemLeft.position) {
                        return itemRight.position - itemLeft.position;
                      } else {
                        return itemLeft.word.length - itemRight.word.length;
                      }
                    });
                  });

                  // merge hits into slices

                  function mergeIntoSlice(text, start, end, index) {
                    var item = index[index.length - 1];
                    var position = item.position;
                    var word = item.word;
                    var hits = [];
                    var searchTextCountInSlice = 0;
                    while (position + word.length <= end && index.length != 0) {
                      if (word === searchText) {
                        searchTextCountInSlice++;
                      }
                      hits.push({position: position, length: word.length});
                      var wordEnd = position + word.length;

                      // move to next position of hit

                      index.pop();
                      while (index.length != 0) {
                        item = index[index.length - 1];
                        position = item.position;
                        word = item.word;
                        if (wordEnd > position) {
                          index.pop();
                        } else {
                          break;
                        }
                      }
                    }
                    searchTextCount += searchTextCountInSlice;
                    return {
                      hits: hits,
                      start: start,
                      end: end,
                      searchTextCount: searchTextCountInSlice
                    };
                  }

                  var slicesOfTitle = [];
                  if (indexOfTitle.length != 0) {
                    slicesOfTitle.push(mergeIntoSlice(title, 0, title.length, indexOfTitle));
                  }

                  var slicesOfContent = [];
                  while (indexOfContent.length != 0) {
                    var item = indexOfContent[indexOfContent.length - 1];
                    var position = item.position;
                    var word = item.word;
                    // cut out 100 characters
                    var start = position - 20;
                    var end = position + 80;
                    if(start < 0){
                      start = 0;
                    }
                    if (end < position + word.length) {
                      end = position + word.length;
                    }
                    if(end > content.length){
                      end = content.length;
                    }
                    slicesOfContent.push(mergeIntoSlice(content, start, end, indexOfContent));
                  }

                  // sort slices in content by search text's count and hits' count

                  slicesOfContent.sort(function (sliceLeft, sliceRight) {
                    if (sliceLeft.searchTextCount !== sliceRight.searchTextCount) {
                      return sliceRight.searchTextCount - sliceLeft.searchTextCount;
                    } else if (sliceLeft.hits.length !== sliceRight.hits.length) {
                      return sliceRight.hits.length - sliceLeft.hits.length;
                    } else {
                      return sliceLeft.start - sliceRight.start;
                    }
                  });

                  // select top N slices in content

                  var upperBound = parseInt('1');
                  if (upperBound >= 0) {
                    slicesOfContent = slicesOfContent.slice(0, upperBound);
                  }

                  // highlight title and content

                  function highlightKeyword(text, slice) {
                    var result = '';
                    var prevEnd = slice.start;
                    slice.hits.forEach(function (hit) {
                      result += text.substring(prevEnd, hit.position);
                      var end = hit.position + hit.length;
                      result += '<b class="search-keyword">' + text.substring(hit.position, end) + '</b>';
                      prevEnd = end;
                    });
                    result += text.substring(prevEnd, slice.end);
                    return result;
                  }

                  var resultItem = '';

                  if (slicesOfTitle.length != 0) {
                    resultItem += "<li><a href='" + articleUrl + "' class='search-result-title'>" + highlightKeyword(title, slicesOfTitle[0]) + "</a>";
                  } else {
                    resultItem += "<li><a href='" + articleUrl + "' class='search-result-title'>" + title + "</a>";
                  }

                  slicesOfContent.forEach(function (slice) {
                    resultItem += "<a href='" + articleUrl + "'>" +
                      "<p class=\"search-result\">" + highlightKeyword(content, slice) +
                      "...</p>" + "</a>";
                  });

                  resultItem += "</li>";
                  resultItems.push({
                    item: resultItem,
                    searchTextCount: searchTextCount,
                    hitCount: hitCount,
                    id: resultItems.length
                  });
                }
              })
            };
            if (keywords.length === 1 && keywords[0] === "") {
              resultContent.innerHTML = '<div id="no-result"><i class="fa fa-search fa-5x" /></div>'
            } else if (resultItems.length === 0) {
              resultContent.innerHTML = '<div id="no-result"><i class="fa fa-frown-o fa-5x" /></div>'
            } else {
              resultItems.sort(function (resultLeft, resultRight) {
                if (resultLeft.searchTextCount !== resultRight.searchTextCount) {
                  return resultRight.searchTextCount - resultLeft.searchTextCount;
                } else if (resultLeft.hitCount !== resultRight.hitCount) {
                  return resultRight.hitCount - resultLeft.hitCount;
                } else {
                  return resultRight.id - resultLeft.id;
                }
              });
              var searchResultList = '<ul class=\"search-result-list\">';
              resultItems.forEach(function (result) {
                searchResultList += result.item;
              })
              searchResultList += "</ul>";
              resultContent.innerHTML = searchResultList;
            }
          }

          if ('auto' === 'auto') {
            input.addEventListener('input', inputEventFunction);
          } else {
            $('.search-icon').click(inputEventFunction);
            input.addEventListener('keypress', function (event) {
              if (event.keyCode === 13) {
                inputEventFunction();
              }
            });
          }

          // remove loading animation
          $(".local-search-pop-overlay").remove();
          $('body').css('overflow', '');

          proceedsearch();
        }
      });
    }

    // handle and trigger popup window;
    $('.popup-trigger').click(function(e) {
      e.stopPropagation();
      if (isfetched === false) {
        searchFunc(path, 'local-search-input', 'local-search-result');
      } else {
        proceedsearch();
      };
    });

    $('.popup-btn-close').click(onPopupClose);
    $('.popup').click(function(e){
      e.stopPropagation();
    });
    $(document).on('keyup', function (event) {
      var shouldDismissSearchPopup = event.which === 27 &&
        $('.search-popup').is(':visible');
      if (shouldDismissSearchPopup) {
        onPopupClose();
      }
    });
  </script>





  

  

  

  

  

  

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</html>
